You
don't need a great deal of imagination to foresee an increasing
significance of lightwave technology in data processing and
telecommunications. Here are some arguments in favor of
light:
Miniaturization of electronic circuits leads to increased
resistances and hence larger dissipation. Photons don't suffer
from losses in the same degree because their interaction is
much weaker than that of electrons. The bandwidths available for
signal transmission are a few hundred kHz on copper cables,
versus roughly a THz in a typical glass fiber - even now it is
feasible to carry half a million telephone conversations over a
single glass fiber. Photons are the method of choice for
massively parallel data processing and storage.

All of us
(physicists) have probably been "exposed" to the He-Ne laser in
some graduate student lab. But of course the most ubiquitous
lasers are by now the semiconductor diode lasers. Both of these
incarnations rely on the parallel-mirror configuration to
provide the feedback that makes laser action possible. This
type of resonator is also known from the Fabry-Perot
interferometer.

One common way of making especially good
parallel mirrors is to use Bragg reflection at multiple layers
of dielectric films. See, e.g., the Wikipedia entry on
"Vertical-Cavity Surface Emitting Lasers". The Bragg
principle is based on the destructive interference
between waves in successive layers of a stack of dielectric
layers.

As a rather logical continuation of the same principle, one
has progressed to photonic
crystals which employ the Bragg principle in more than one
spatial direction and can in principle be used to make
extremely small photonic cavities. The price one pays is that
one needs many periods of the artificial crystal lattice in
order to obtain high reflectivities, so that the total size of
the structure ends up being much larger than the cavity itself.
Higher and higher reflectivities are required, on the other
hand, if one wants to make a laser out of such a
microcavity. The simple reason is that a small cavity can host
only a small amount of amplifying material, and therefore it
becomes more difficult for amplification to win over the losses
in a microcavity laser.

In solid-state laser materials, it
is often possible to realize the mirrors simply by exploiting
total internal reflection at the interface between the
high-index solid and the surrounding medium (e.g., air). In
contrast to the Bragg principle, this confinement mechanism for
light is to lowest order frequency-independent and can
therefore be called a classical effect
- it can be described without explicit use of the wave nature
of light, by using Fermat's principle.
This is good because it means that a device based on this
confinement mechanism will in principle be able to work over a
very broad range of wavelengths - in stark contrast to photonic
crystals. Nevertheless, one can use total internal reflection
to make three-dimensionally confined resonators with
high frequency selectivity (or "finesse"), provided
one can force wavefronts inside the cavity to interfere with
themselves.

This is achieved with the "whispering-gallery"
resonator which is at the heart of the lowest-threshold lasers
made so far. This low threshold becomes possible as a
consequence of the small size that can be achieved with these
resonators. They are essentially circular disks in which the
light circulates around close to the dielectric interface. Such
modes are especially low in losses.

Whispering-gallery
waves:
To illustrate the whispering-gallery effect, the movie
shows a cross sectional view of a curved interface (black
circle) between glass and air, with a circulating wave
radiating in all directions.
The color represents the electric field, and in the first
animation the field inside the resonator is only slightly
higher than outside. This is not a good resonator because
it is very "lossy".
In the second movie, the wavelength is about 4 times
shorter than above. In this case, the field outside the
resonator is much weaker than inside it, meaning that we
are confining the light much better. In both animations,
the wave fronts look slanted, especially on the
outside. Comparing the two clips, you will notice, however,
that the wave fronts right at the circular
interface are perfectly radial in the bottom image.
This is what makes the two scenarios different: the
straight wave fronts at the interface correspond to grazing
propagation along the curved boundary.
There is still a wave emanating from the cavity at the
bottom, but its amplitude relative to that at the interface
is now much smaller. Observe also the central region of the
dielectric circle, which is essentially field-free. The
intensity is highly concentrated near the surface.

Even in the more strongly confined case, shown here, the wave penetrates
slightly into the surrounding medium. In reflection off a
straight dielectric interface, this penetration is known to go along with
the
Goos-Hänchen effect, a lateral
displacement of the scattered beam. A calculation of the analogous effect
in reflection off a curved interface can be done starting from the
circular geometry. Since the Goos-Hänchen effect can be incorporated
into a ray model, it improves semiclassical calculations for non-circular
cavities.

Semiconductors are far from being the only application of
the whispering-gallery mechanism. The first laser resonators in
the submillimeter size regime were made of liquid droplets containing a lasing organic
dye. The highest-quality optical microresonators have been
achieved using fused-silica spheres (i.e., glass). Although
these materials have a refractive index closer to unity than a
semiconductor, they still support whispering-gallery modes. In
that context, they are often called morphology-dependent
resonances (MDRs).

Both the semiconductor and the droplet realizations of the
whispering gallery are illustrated on the cover of

Optical Processes in Microcavities,
edited by R.K.Chang and A.J.Campillo (World Scientific
Publishers, 1996).
The lasing droplets are seen on the left side, and a
"thumbtack" microlaser with its rotationally symmetric
calculated emission pattern appears in the main panel.

Remark:
This book contains 11 chapters on important experimental and
theoretical aspects of dielectric microcavities. Chapter 11
represents the status of our work as of summer 1995:"Chaotic Light: a theory of asymmetric
cavity resonators",
J.U.Nöckel and A.D.StonePDF - (warning:
large files)

The question that arises naturally in lasing microdroplets
is: how strongly can a dielectric resonator be deformed
before whispering-gallery modes cease to exist, or become
degraded by leakage? The intuitive answer is, "the rounder, the
better". However, even shapes with sharp corners can sustain
modes that have every right to be called whispering-gallery
phenomena. In fact, these types of whispering-gallery modes
cannot be understood purely on the basis of ray optics. This is
discussed in our work on hexagonal
nanoporous microlasers.

Being round is not a prerequisite for whispering-gallery
action. So there is a huge space of possible shapes
(practically from circle to square) that could possibly be
considered as whispering-gallery type resonators. If we had a
choice, what should the ideal shape be? This clearly depends on
the application context, but in any case it would be desirable
to have some design rules. In the following, we begin to
discuss some design issues, and point out how our work in
particular aims to provide the design rules just mentioned,
based on approximate methods such as the ray picture.

Other mirror arrangements provide different
advantages. In particular, there has been a considerable body
of work employing concave or convex mirrors. E.g., concave
mirrors separated by less than their radii of curvature added
together, make a stable resonator in which light rays undergo
focussing while being multiply reflected between the mirrors.
Light can then be coupled out by making one of the mirrors
slightly transparent.
When the output coupling is small, the theoretical treatment of
such a laser can often be performed by neglecting the leakage
and hence assuming the existence of some orthogonal set of
modal eigenfunctions.

If one wants to avoid the use of partially transparent
mirrors (which need to have very low losses for high-power
applications), one alternative design is the unstable
resonator containing defocussing elements [see the
exhaustive textbook by A.E.Siegman, Lasers
(University Science Books, Mill Valley, CA (1986)].
E.g., two concave mirrors separated by more than their added
radii of curvature cause rays to diverge out from the optical
axis after several reflections. Outcoupling occurs when the
light spills over the edge of one of the mirrors (which hence
need not be partially transparent themselves).

Such unstable lasers differ from stable resonators in their
mode structure: A set of well-defined bound modes is not
available for the expansion of the laser field, because they
all couple to the outside. Therefore, it has been necessary to
use quasibound states in the calculations.

Lasers are fundamentally open systems, so a description in
terms of quasibound states seems only natural. These states
are, however, not as familiar a tool as the usual
square-integrable eigenfunctions one knows from bound systems.
Their properties are still a topic of current research.

Remark:
Important work on such "quasi-normal modes" has also been
carried out by Kenneth Young, Pui-Tang Leung and co-workers.
The central problem from the point of view of laser physics
is this:
In order to define photons in the first place, we expect to
have at our disposal a set of normal modes for which we then
write the creation and annihilation operators. But metastable
states are not eigenstates of a Hermitian differential
operator, because they represent energy escaping to infinity.
Therefore, familiar precedures involving expansions in normal
modes run into problems.

Nevertheless, their use makes a lot of sense when
discussing the emission properties of individual
metastable states, such as their frequency shifts as a result
of a perturbation in the resonator's shape or dielectric
constant.

Or - just to mention a really far-out example: metastable states find
application in the study of gravitational waves emitted from
a black hole [P.T.Leung et al.,
Phys.Rev.Lett. 78, 2894 (1997)]

As an
extention of the unstable-resonator idea, one can think of two
concave mirrors in a defocussing setup combined with some
lateral (sideways) guiding of the light between the mirrors. A
naive reasoning could be this:
We want lasing from light spilling out near one of the mirrors,
but we don't want the escape angle with the optical axis to be
too large, hoping thereby to improve the spatial mode pattern
(focussing). So we put additional mirrors along the open sides
joining the mirrors.

Now combine this idea with the use of dielectric interfaces
as (partially transparent) mirrors, and one is lead quite
directly to consider the so-called stadium resonator (or a
generalization thereof).

Here is an illustration of the stadium shape and of how it
scatters an incident ray:

It is taken from J.H.Jensen, J.Opt.Soc.Am.A 10 (1993).

Remark on previous work:
Jensen seems to have been the first to attack the ray-wave
duality for a stadium-shaped dielectric resonator, in
particular taking into account the inevitable
ray-splitting into reflected and transmitted portions
that occurs at the sharp dielectric interface of the chaotic
resonator (thanks to R.K. Chang and A. Poon for pointing out
the reference). However, he did not consider the long-lived
resonances that such a cavity could support, which are a
prerequisite for lasing. Instead, Jensen's paper gives a
quasiclassical analysis of the rainbow-peaks for this
structure. For more on rainbows, see this Atmospheric Optics web
site. Ray splitting has received renewed interest in
recent years (in my own ray optics simulations, it is taken
into account as well - it becomes essential in high-index
materials).

We are not the only ones to consider chaotic dielectric
resonators. However, we were the first (to my knowledge) to
seriously apply chaos analysis to the emission properties of
quasibound states in dielectric resonators, see"Q spoiling and directionality in deformed
ring cavities",
J.U.Nöckel, A.D.Stone and R.K.Chang, Optics Letters
19, 1693 (1994).PDF.
This is a theory paper in which we address the consequences
of emerging ray chaos for the lifetimes and emission
directionality of deformed dielectric resonators.

The first experiment in which the correspondence
between emission anisotropy and chaotic structure in the
classical ray dynamics was successfully applied to dielectric
microlasers is"Ray chaos and Q-spoiling in lasing
droplets",
A.Mekis, J.U.Nöckel, G.Chen, A.D.Stone and R.K.Chang,
Phys.Rev.Lett. 75, 2682 (1995).PDF.
In this paper, we studied lasing microdroplets with a
nonspherical shape, which leads to a strongly anisotropic
light output along the droplet surface. The total-intensity
profile was imaged and compared with a ray model, yielding an
explanation for the observed features.

To arrive at the idea of using a chaotic resonator cavity,
one can either start from the unstable-resonator concept as
described above, or from the whispering-gallery
design. We came from the latter direction. The argument leading
to an oval dielectric resonator is simply that a circular
whispering-gallery cavity does not have a preferred emission
direction, owing to its rotational symmetry. In addition, one
wishes to have a parameter with which the resonance lifetimes
of the cavity can be controlled. This is achieved by deforming
its shape.

Inbetween
stable and unstable resonators, there is another useful mirror
configuration, called confocal. It has the advantage of
creating a focussing effect inside the resonator, which in turn
amounts to producing a smaller effective mode volume for the
laser. Instead of the whole volume between the mirrors, it is
possible to utilize only a smaller volume around the coinciding
focal points of the mirrors. The ray pattern that forms in a
confocal arrangement of two concave mirrors can sometimes take
on the shape of a bowtie (depending on the shape of the
mirrors). This well-known configuration is found in etalons but
also in lasers. The simplest confocal cavity would consist of
two circle segments with a common focus. A less trivial example
is the case of two confocal paraboloids, i.e., surfaces
of revolution generated by opposing parabolas that share their
focal point:
The righthand picture shows two bowtie rays going through the
focus. There are many other ray paths that never go through the
focus, but they form caustics which are reminiscent of this
basic shape. For a study if this type of (three-dimensional)
mirror configuration, see
my work with Izo Abram's group at CNET, "Mode
structure and ray dynamics of a parabolic dome
microcavity". This is the manuscript:PDF.

Microresonators such as this can find application in
quantum
electrodynamics because they allow to modify the rate of
spontaneous emission of atoms or quantum dots interacting with
the electromagnetic field. To that end, one has to go to small
mode volumes. But the cavity volume isn't necessarily
what counts. With a focused ray pattern as in the
confocal resonator, the light field is especially strong in
only certain portions of the resonator, notably the focal point
in the center. And that is where the desired strong coupling
between the light and the active medium occurs.

The microcylinder laser shown here is not circular, but not
a stadium shape, either. The stadium has fully chaotic ray
dynamics, the circle has no chaos at all. This oval shape has a
mixed phase space. As a by-product of the transition to chaos
which takes place with increasing deformation, a bowtie-shaped
ray path is born that does not exist below a certain
eccentricity. This pattern combines internal and external
focussing, and its lifetime is long enough for lasing because
the rays hit the surface close to the critical angle for total
internal reflection.

This is the world's most powerful
microlaser to date.
To understand why this very desirable intensity distribution
arises in the smooth oval shape we chose here, but not in the
circle or the stadium, one has to use methods of classical
nonlinear dynamics. This is explained in our article," High power directional emission from lasers
with chaotic resonators ",
C.Gmachl, F.Capasso, E.E.Narimanov, J.U.Nöckel A.D.Stone,
J.Faist, D.Sivco and A.Cho, Science 280, 1556 (1998)PDF, cond-mat/9806183.

Remark:
In this paper, the oval-resonator concept is combined with a
very innovative laser material that turns out to be
particularly compatible with a disk-shaped resonator
geometry: the quantum cascade laser.

This active material consists of a semiconductor
heterostructure in which an electrical current leads to the
emission of photons. But in contrast to more conventional
quantum-well diode lasers, the optical transitions
responsible for the creation of the photons take place
exclusively within the nanostructured conduction band
(between quantum well subbands). Electron-hole recombination
across the valence band (the usual mechanism) is not involved
here, leading to various advantages.

F.Capasso and J.Faist are among the winners of the 1998
Rank Prize for the invention of the quantum cascade
laser.

We are
talking here about deterministic chaos. The term refers
to the fact that even simple classical systems governed by
simple equations such as Newton's laws can exhibit highly
irregular motion that defies long-term predictions. One example
for such a simple physical system is the double
pendulum; as the following animation shows, the two degrees
of freedom represented by the two angles θ and ψ are
coupled, and this leads to a non-periodic,
unpredictable-looking combined motion:

In Optics, there is a slight
confusion of terminology about the concept of chaos, because it
is traditionally found (in quantum optics) when people want to
describe the statistical properties of a photon source.
"Chaotic light" in that context has a much shallower meaning -
it just means "random" thermal distribution of photons as it is
found in blackbody radiation.

Chaos in the deterministic sense already has a
place in optics as well, but again we have to make a
distinction to our work. In multimode lasing one can look at
the temporal and/or spatial evolution of the laser emission and
finds that the signal can become very irregular. By mapping
this behavior onto an artificial (usually many-dimensional)
space, e.g. by a so-called time-delay embedding, one then
sometimes finds that the system follows a trajectory on a
"chaotic attractor". That's a type of structure one finds in
dissipative
nonlinear classical systems. This is what people have
studied in nonlinear optics for a long time now.

There are many lists of chaos-science links; see for example
the Wikipedia
artticle on this subject. For more on the the relation
between our work and the more traditional nonlinear optics, see
below.

In the
classical ray picture for our microresonators, the fact that
boundaries are penetrable does not (to lowest order in the
wavelength) affect the shape of the trajectories, and hence our
internal ray dynamics is that of a non-dissipative,
closed system.

If you have any further questions about chaos, you may well
find an answer at this
informative FAQ site maintained by Jim Meiss. Further
information, including a host of graphics and animations, is
also available from the chaos group at the University of
Maryland.

Quantum chaos sounds like a contradiction in
terms because linear wave equations such as the
Schrödinger equation do not exhibit the sensitivity to
initial conditions that gives rise to chaos. Nonetheless,
classical mechanics is just a limiting case of quantum
mechanics, just as ray optics is the limit of wave optics for
short wavelengths. So one should expect "signatures of chaos"
in the wave solutions. To find and understand these,
semiclassical methods are indispensable.

One of the pioneers of quantum chaos, Martin C.
Gutzwiller, has written a beautiful introduction
to this field in Scientific American. See in particular the
third figure describing the central place of quantum chaos in
our our understanding of quantum mechanics. An important lesson
here is: Playing around with the simple standard systems,
such as harmonic oscillators, we barely scratch the surface of
what the classical-quantum transition really entails. If we
want to go beyond pedestrian descriptions of this transition,
classically chaotic systems are where the action is! This
also holds for much-discussed fundamental topics such as
"decoherence", see the example of periodically
"kicked" Cesium atom. As a by-product, quantum chaos has
brought together an arsenal of powerful techniques. My first
chance to study these was a graduate course at Yale taught by
Prof. Gutzwiller in 1993/94; he also accompanied my thesis work
on chaotic optical cavities through discussions and as a reader
at dissertation time.

As it turns out, many of the intrinsic emission properties
of dielectric optical resonators have a classical origin. The
significance of this for quantum chaos is that comparison
between ray model and numerical solutions of the wave equations
uncover corrections to the ray model. Alternatively, one
can also discover such wave corrections by comparing the ray
predictions to an actual experiment. We follow both
approaches.

Such wave corrections become especially interesting when the
underlying classical dynamics is partially chaotic, as
is the case in the asymmetric dielectric resonators. In that
setting, two major new effects arise:dynamical localization and dynamical
tunneling.

In dielectric cavities, the effect of such phenomena on
resonance lifetimes and emission directionality, and of course on resonance
frequencies, can be studied. Emission directionality is in
itself a completely new question to investigate from the
viewpoint of quantum chaos: when decay occurs, e.g.,in nuclear
physics or chemistry, any anisotropy of the individual process
is averaged out in the observation of an ensemble - but
microlasers can be looked at individually, and from various
directions. If they are bounded only by a dielectric interface,
the emission pattern is determined by the phase-space
structure. This is an important focus of my work: the
short-wavelength asymptotics of systems that are chaotic
and open.

What this means is illustrated in a slightly different
example on a picture page describing the annular billiard. There, we
studied the relation between resonance lifetimes and dynamical
tunneling (since it involves tunneling into a chaotic portion
of phase space, it is also called "chaos-assisted
tunneling").

Is quantum chaos just a mathematical-conceptual game without
relevance for experiments? Our work has been among the first to
propose actual applications of quantum chaos phenomena, and to
my knowledge the two patents I co-authored were the very first
to rely on such phenomena.

Nonlinear dynamics

Chaos, belonging to the field of
nonlinear dynamics, is known to laser physicists in another
guise as well: pattern formation, in particular vortices and
vortex lattices, due to the nonlinearity of the lasing medium,
has been studied much longer than our type of chaotic phenomena
which rely on the boundary effects. Of course, there can be a
cross-over from one regime to the other, e.g. from nonlinear
vortices to linear vortices which in a circular resonator are
encountered as whispering-gallery modes.

What I'm discussing above is
chaos in the linear wave equation. This phenomenon often
dominates the physics, especially near the lasing threshold. At higher
powers the nonlinearity of the medium itself becomes more
important.

Last significant revision: 09/09/04.
This page represents a compilation of information
relevant to our work on microlaser resonators. Naturally,
it cannot claim to be complete in any way. However, I
felt it appropriate to provide some context because the
questions we are discussing are at the interface between
two fields of study that traditionally haven't had much
overlap: micro-optics and quantum chaos.

These fields have more in common than meets
the eye.
But that by no means implies that one community cannot
learn from the other...

Since this is a NET DOCUMENT, I am trying
to refer mostly to other documents that are available
online, instead of citing things printed on dead trees.
But if you have something you'd like me to include, feel
free to let me know.